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I live in Mexico City which is one of the most densely populated cities in the world. I'd say 70-90% of clothes are washed by hand here still including by myself occasionally, and yes still using clotheslines. I really don't understand the backlash to my comment. People manage fine here by not missing school because they don't have washing machines.
I think there's a bit of disconnect between your point and the reaction to it. The fact is that people have long been able to do laundry without more expensive resources. It doesn't matter if you're in Mexico City or Appalachia or a Chicago housing project . . . people can do the laundry.
So perhaps you're right that not having clean clothes is merely a symptom of broader failures that are going on in our culture. That said, if putting in some laundry machines demonstrably keeps kids in school, it sounds like a meaningful endeavor.